Xuepeng Li and Wensheng Yang

Copyright © 2010 Xuepeng Li and Wensheng Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We propose a discrete model of mutualism with infinite deviating arguments, that is x1(n+1)=x1(n)exp{r1(n)[(K1(n)+α1(n)∑s=0∞J2(s)x2(n−s))/(1+∑s=0∞J2(s)x2(n−s))−x1(n−σ1(n))]},x2(n+1)=x2(n)exp{r2(n)[(K2(n)+α2(n)∑s=0∞J1(s)x1(n−s))/(1+∑s=0∞J1(s)x1(n−s))−x2(n−σ2(n))]}. By some Lemmas, sufficient conditions are obtained for the permanence of the system.

1. Introduction

Chen and You [1] studied the following two species integro-differential model of mutualism:

(1.1) where , and are continuous functions bounded above and below by positive constants: and Using the differential inequality theory, they obtained a set of sufficient conditions to ensure the permanence of system (1.1). For more background and biological adjustments of system(1.1), one could refer to [1–4] and the references cited therein.

However, many authors [5–12] have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Also, since discrete time models can also provide efficient computational models of continuous models for numerical simulations, it is reasonable to study discrete time models governed by difference equations. Another permanence is one of the most important topics on the study of population dynamics. One of the most interesting questions in mathematical biology concerns the survival of species in ecological models. It is reasonable to ask for conditions under which the system is permanent.

Motivated by the above question, we consider the permanence of the following discrete model of mutualism with infinite deviating arguments:

(1.2) where is the density of mutualism species at the th generation. For , and are bounded nonnegative sequences such that

(1.3) Here, for any bounded sequence

Let we consider (1.2) together with the following initial condition:

(1.4)

It is not difficult to see that solutions of (1.2) and (1.4) are well defined for all and satisfy

(1.5)

The aim of this paper is, by applying the comparison theorem of difference equation and some lemmas, to obtain a set of sufficient conditions which guarantee the permanence of system (1.2).

2. Permanence

In this section, we establish permanence results for system (1.2).

Following Comparison Theorem of difference equation is Theorem 2.6 of [13, page ].

Lemma. Let . For any fixed is a non-decreasing function with respect to , and for , following inequalities hold: If , then for all .

Now let us consider the following single species discrete model:

(2.1) where and are strictly positive sequences of real numbers defined for and . Similar to the proof of Propositions and in [6], we can obtain the following.

Lemma. Any solution of system (2.1) with initial condition satisfies (2.2) where (2.3)

Lemma 2.3 (see [14]). Let and be nonnegative sequences defined on , and is a constant. If (2.4) then (2.5)

Lemma 2.4 (see [2]). Let be a nonnegative bounded sequences, and let be a nonnegative sequence such that . Then (2.6)

Proposition. Let be any positive solution of system (1.2), then (2.7) where (2.8)

Proof. Let be any positive solution of system (1.2), then from the first equation of system (1.2) we have (2.9) Let , then (2.10) where (2.11) When is nonnegative sequence, by applying Lemma 2.3, it immediately follows that (2.12) When is negative sequence, (2.12) also holds. From (2.12), we have (2.13) By using the second equation of system (1.2), similar to the above analysis, we can obtain (2.14) This completes the proof of Proposition 2.5.

Now we are in the position of stating the permanence of system (1.2).

Theorem. Under the assumption(1.3), system (1.2) is permanent, that is, there exist positive constants which are independent of the solutions of system (1.2) such that, for any positive solution of system(1.2) with initial condition (1.4), one has (2.15)

Proof. By applying Proposition 2.5, we see that to end the proof of Theorem 2.6 it is enough to show that under the conditions of Theorem 2.6(2.16) From Proposition 2.5, For all , there exists a For all (2.17) According to Lemma 2.4, from (2.13) and (2.14) we have (2.18) For above , according to (2.18), there exists a positive integer , such that, for all (2.19) Thus, for all , from the first equation of system(1.2), it follows that (2.20) It follows that, for (2.21) Hence (2.22) In other words, (2.23) From the first equation of system (1.2) and (2.23), for all , it follows that (2.24) By applying Lemmas 2.1 and 2.2 to (2.24), it immediately follows that(2.25) Setting , it follows that (2.26) Similar to the above analysis, from the second equation of system (1.2), we have that (2.27) This completes the proof of Theorem 2.6.

References

1. F. D. Chen and M. S. You, “Permanence for an integrodifferential model of mutualism,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 30–34, 2007.
2. F. D. Chen, “Permanence in a discrete Lotka-Volterra competition model with deviating arguments,” Nonlinear Analysis: Real World Applications, vol. 9, no. 5, pp. 2150–2155, 2008.
3. Y. K. Li and G. T. Xu, “Positive periodic solutions for an integrodifferential model of mutualism,” Applied Mathematics Letters, vol. 14, no. 5, pp. 525–530, 2001.
4. P. Yang and R. Xu, “Global asymptotic stability of periodic solution in n-species cooperative system with time delays,” Journal of Biomathematics, vol. 13, no. 6, pp. 841–846, 1998.
5. R. P. Agarwal, Difference Equations and Inequalities: Theory, Method and Applications, vol. 228 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000.
6. F. D. Chen, “Permanence and global attractivity of a discrete multispecies Lotka-Volterra competition predator-prey systems,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 3–12, 2006.
7. X. Chen and F. D. Chen, “Stable periodic solution of a discrete periodic Lotka-Volterra competition system with a feedback control,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 1446–1454, 2006.
8. Y. K. Li and L. H. Lu, “Positive periodic solutions of discrete n-species food-chain systems,” Applied Mathematics and Computation, vol. 167, no. 1, pp. 324–344, 2005.
9. Y. Muroya, “Persistence and global stability in Lotka-Volterra delay differential systems,” Applied Mathematics Letters, vol. 17, no. 7, pp. 795–800, 2004.
10. Y. Muroya, “Partial survival and extinction of species in discrete nonautonomous Lotka-Volterra systems,” Tokyo Journal of Mathematics, vol. 28, no. 1, pp. 189–200, 2005.
11. X. T. Yang, “Uniform persistence and periodic solutions for a discrete predator-prey system with delays,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 161–177, 2006.
12. F. D. Chen, “Permanence for the discrete mutualism model with time delays,” Mathematical and Computer Modelling, vol. 47, no. 3-4, pp. 431–435, 2008.
13. L. Wang and M. Q. Wang, Ordinary Difference Equation, Xinjiang University Press, Xinjiang, China, 1991.
14. Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems, World Scientific, River Edge, NJ, USA, 1996.